Closed-loop analysis for systems with fast linear dynamics preceded by hysteresis

Piezoelectric actuators are commonly modelled by a hysteresis operator preceding fast, stable linear dynamics. This motivates our work to analyze systems with these characteristics when a popular control architecture involving both hysteresis inversion and feedback is adopted. In particular, we are interested in the frequency-scaling behavior of the tracking error for such systems, which is of practical interest but has received little attention in the literature. The hysteresis nonlinearity in our analysis is represented by piecewise linear segments, which is applicable to many hysteresis operators. To fix ideas, we consider a proportional integral controller for the feedback component, as well as the case where a constant-gain feedforward component is added to the feedback term. This work is a continuation of our previous work where we only examined the system behavior for a given hysteresis segment. Here we use singular perturbation techniques to separate the slow variables of the controller from the fast variables of the plant dynamics, and derive the solution of the closed-loop system and the tracking error at the steady state under a sinusoidal reference. The analysis incorporates the effect of uncertainty in the hysteresis model, and offers insight into how the tracking error scales with the reference frequency. The analysis is confirmed with experimental and simulation results for the control of a piezo-actuated nanopositioner.